L
The Bell System Technical Journal
October, 1933
Loudness, Its Definition, Measurement and Calculation*
By HARVEY FLETCHER and W. A. MUNSON
An empirical formula for calculating the loudness of any steady sound
from an analysis of the intensity and frequency of its components is devel-
oped in this article. The development is based on fundamental properties
of the hearing mechanism in such a way that a scale of loudness values
results. In order to determine the form of the function representing this
loudness scale and of the other factors entering into the loudness formula,
measurements were made of the loudness levels of many sounds, both of pure
tones and of complex wave forms. These tests are described and the method
of measuring loudness levels is discussed in detail. Definitions are given
endeavoring to clarify the terms used and the measurement of the physical
quantities which determine the characteristics of a sound wave stimulating
the auditory mechanism.
Introduction
OUDNESS is a psychological term used to describe the magnitude
of an auditory sensation. Although we use the terms "very
loud," "loud," "moderately loud," "soft" and "very soft," corre-
sponding to the musical notations/, /, mf, p, and pp, to define the
magnitude, it is evident that these terms are not at all precise and
depend upon the experience, the auditory acuity, and the customs of
the persons using them. If loudness depended only upon the intensity
of the sound wave producing the loudness, then measurements of the
physical intensity would definitely determine the loudness as sensed
by a typical individual and therefore could be used as a precise means
of defining it. However, no such simple relation exists.
The magnitude of an auditory sensation, that is, the loudness of the
sound, is probably dependent upon the total number of nerve impulses
that reach the brain per second along the auditory tract. It is evident
that these auditory phenomena are dependent not alone upon the in-
tensity of the sound but also upon their physical composition. For
example, if a person listened to a flute and then to a bass drum placed
at such distances that the sounds coming from the two instruments
are judged to be equally loud, then the intensity of the sound at the ear
produced by the bass drum would be many times that produced by
the flute.
If the composition of the sound, that is, its wave form, is held con-
stant, but its intensity at the ear of the listener varied, then the loud-
* Jour. Acous. Soc. Amer., October, 1933.
377
-
378 BELL SYSTEM TECHNICAL JOURNAL
ness produced will be the same for the same intensity only if the same
or an equivalent ear is receiving the sound and also only if the listener
is in the same psychological and physiological conditions, with reference
to fatigue, attention, alertness, etc. Therefore, in order to determine
the loudness produced, it is necessary to define the intensity of the
sound, its physical composition, the kind of ear receiving it, and the
physiological and psychological conditions of the listener. In most
engineering problems we are interested mainly in the effect upon a
typical observer who is in a typical condition for listening.
In a paper during 1921 one of us suggested using the number of
decibels above threshold as a measure of loudness and some experi-
mental data were presented on this basis. As more data were accumu-
lated it was evident that such a basis for defining loudness must be
abandoned.
In 1924 in a paper by Steinberg and Fletcher l some data were
given which showed the effects of eliminating certain frequency bands
upon the loudness of the sound. By using such data as a basis, a
mathematical formula was given for calculating the loudness losses of
a sound being transmitted to the ear, due to changes in the trans-
mission system. The formula was limited in its application to the
particular sounds studied, namely, speech and a sound which was
generated by an electrical buzzer and called the test tone.
In 1925 Steinberg 2 developed a formula for calculating the loudness
of any complex sound. The results computed by this formula agreed
with the data which were then available. However, as more data
have accumulated it has been found to be inadequate. Since that
time considerably more information concerning the mechanism of
hearing has been discovered and the technique in making loudness
measurements has advanced. Also more powerful methods for pro-
ducing complex tones of any known composition are now available.
For these reasons and because of the demand for a loudness formula of
general application, especially in connection with noise measurements,
the whole subject was reviewed by the Bell Telephone Laboratories and
the work reported in the present paper undertaken. This work has
resulted in better experimental methods for determining the loudness
level of any sustained complex sound and a formula which gives
calculated results in agreement with the great variety of loudness data
which are now available.
1 H. Fletcher and J. C. Steinberg, " Loudness of a Complex Sound," Phys, Rev. 24,
306(1924).
2 J. C. Steinberg, "The Loudness of a Sound and Its Physical Stimulus," Phys. Rev.
26, 507 (1925).
LOUDNESS 379
Definitions
The subject matter which follows necessitates the use of a number of
terms which have often been applied in very inexact ways in the past.
Because of the increase in interest and activity in this field, it became
desirable to obtain a general agreement concerning the meaning of the
terms which are most frequently used. The following definitions are
taken from recent proposals of the sectional committee on Acoustical
Measurements and Terminology of the American Standards Associ-
ation and the terms have been used with these meanings throughout
the paper.
Sound Intensity
The sound intensity of a sound field in a specified direction at a
point is the sound energy transmitted per unit of time in the specified
direction through a unit area normal to this direction at the point.
In the case of a plane or spherical free progressive wave having the-
effective sound pressure P (bars), the velocity of propagation c (cm.
per sec.) in a medium of density p (grams per cubic cm.), the intensity
in the direction of propagation is given by
J = P2lpc (ergs per sec. per sq. cm.). (1)
This same relation can often be used in practice with sufficient accuracy
to calculate the intensity at a point near the source with only a pressure
measurement. In more complicated sound fields the results given by
this relation may differ greatly from the actual intensity.
When dealing with a plane or a spherical progressive wave it will
be understood that the intensity is taken in the direction of propagation
of the wave.
Reference Intensity
The reference intensity for intensity level comparisons shall be 10-16
watts per square centimeter. In a plane or spherical progressive
sound wave in air, this intensity corresponds to a root-mean-square
pressure p given by the formula
p = 0.000207[(i7/76)(273/r)i]5 (2)
where p is expressed in bars, II is the height of the barometer in
centimeters, and T is the absolute temperature. At a temperature of
20° C. and a pressure of 76 cm. of Hg, p = 0.000204 bar.
Intensity Level
The intensity level of a sound is the number of db above the reference
intensity.
380 BELL SYSTEM TECHNICAL JOURNAL
Reference Tone
A plane or spherical sound wave having only a single frequency of
1,000 cycles per second shall be used as the reference for loudness
comparisons.
Note: One practical way to obtain a plane or spherical wave is to
use a small source, and to have the head of the observer at least one
meter distant from the source, with the external conditions such that
reflected waves are negligible as compared with the original wave at
the head of the observer.
Loudness Level
The loudness level of any sound shall be the intensity level of the
equally loud reference tone at the position where the listener's head is
to be placed.
Manner of Listening to the Sound
In observing the loudness of the reference sound, the observer shall
face the source, which should be small, and listen with both ears at a
position so that the distance from the source to a line joining the two
ears is one meter.
The value of the intensity level of the equally loud reference sound
depends upon the manner of listening to the unknown sound and also
to the standard of reference. The manner of listening to the unknown
sound may be considered as part of the characteristics of that sound.
The manner of listening to the reference sound is as specified above.
Loudness has been briefly defined as the magnitude of an auditory
sensation, and more will be said about this later, but it will be seen
from the above definitions that the loudness level of any sound is ob-
tained by adjusting the intensity level of the reference tone until it
sounds equally loud as judged by a typical listener. The only way of
determining a typical listener is to use a number of observers who have
normal hearing to make the judgment tests. The typical listener, as
used in this sense, would then give the same results as the average
obtained by a large number of such observers.
A pure tone having a frequency of 1000 cycles per second was chosen
for the reference tone for the following reasons: (1) it is simple to
define, (2) it is sometimes used as a standard of reference for pitch,
(3) its use makes the mathematical formulae more simple, (4) its range
of auditory intensities (from the threshold of hearing to the threshold
of feeling) is as large and usually larger than for any other type of
sound, and (5) its frequency is in the mid-range of audible frequencies.
There has been considerable discussion concerning the choice of the
LOUDNESS 381
reference or zero for loudness levels. In many ways the threshold of
hearing intensity for a 1000-cycle tone seems a logical choice. How-
ever, variations in this threshold intensity arise depending upon the
individual, his age, the manner of listening, the method of presenting
the tone to the listener, etc. For this reason no attempt was made to
choose the reference intensity as equal to the average threshold of a
given group listening in a prescribed way. Rather, an intensity of the
reference tone in air of 10~16 watts per square centimeter was chosen
as the reference intensity because it was a simple number which was
convenient as a reference for computation work, and at the same
time it is in the range of threshold measurements obtained when
listening in the standard method described above. This reference
intensity corresponds to the threshold intensity of an observer who
might be designated a reference observer. An examination of a large
series of measurements on the threshold of hearing indicates that such
a reference observer has a hearing which is slightly more acute than
the average of a large group. For those who have been thinking in
terms of microwatts it is easy to remember that this reference level is
100 db below one microwatt per square centimeter. When using these
definitions the intensity level /3, of the reference tone is the same as its
loudness level L and is given by
0r = L = 10 log Jr + 100, (3)
where Jr is its sound intensity in microwatts per square centimeter.
The intensity level of any other sound is given by
0 = 10 log / + 100, (4)
where J is its sound intensity, but the loudness level of such a sound
is a complicated function of the intensities and frequencies of its
components. However, it will be seen from the experimental data
given later that for a considerable range of frequencies and intensities
the intensity level and loudness level for pure tones are approximately
equal.
With the reference levels adopted here, all values of loudness level
which are positive indicate a sound which can be heard by the reference
observer and those which are negative indicate a sound which cannot
be heard by such an observer.
It is frequently more convenient to use two matched head receivers
for introducing the reference tone into the two ears. This can be done
provided they are calibrated against the condition described above.
This consists in finding by a series of listening tests by a number of
382 BELL SYSTEM TECHNICAL JOURNAL
observers the electrical power W\ in the receivers which produces the
same loudness as a level /3i of the reference tone. The intensity level
/3r of an open air reference tone equivalent to that produced in the
receiver for any other power Wr in the receivers is then given by
fir = fil + 10 log (Wr/Wi). (5)
Or, since the intensity level fir of the reference tone is its loudness
level L, we have
L = 10 log Wr + Cr, (6)
where Cr is a constant of the receivers.
In determining loudness levels by comparison with a reference tone
there are two general classes of sound for which measurements are
desired: (1) those which are steady, such as a musical tone, or the hum
from machinery, (2) those which are varying in loudness such as the
noise from the street, conversational speech, music, etc. In this
paper we have confined our discussion to sources which are steady and
the method of specifying such sources will now be given.
A steady sound can be represented by a finite number of pure tones
called components. Since changes in phase produce only second order
effects upon the loudness level it is only necessary to specify the
magnitude and frequency of the components.3 The magnitudes of
the components at the listening position where the loudness level is
desired are given by the intensity levels j3i, /32, • • • fih, ■ ■ • fin of each
component at that position. In case the sound is conducted to the
ears by telephone receivers or tubes, then a value Wk for each com-
ponent must be known such that if this component were acting
separately it would produce the same loudness for typical observers as
a tone of the same pitch coming from a source at one meter's distance
and producing an intensity level of fik-
In addition to the frequency and magnitude of the components of
a sound it is necessary to know the position and orientation of the head
with respect to the source, and also whether one or two ears are used
in listening. The monaural type of listening is important in telephone
use and the binaural type when listening directly to a sound source in
air. Unless otherwise stated, the discussion and data which follow
apply to the condition where the listener faces the source and uses
both ears, or uses head telephone receivers which produce an equivalent
result.
3 Recent work by Chapin and Firestone indicates that at very high levels these
second order effects become large and cannot be neglected. K. E. Chapin and F. A.
Firestone, "Interference of Subjective Harmonics," Jour. Acous. Soc. Am. 4, 176A
(1933).
loudness 383
Formulation of the Empirical Theory for Calculating the
Loudness Level of a Steady Complex Tone
It is well known that the intensity of a complex tone is the sum of
the intensities of the individual components. , Similarly, in finding a
method of calculating the loudness level of a complex tone one would
naturally try to find numbers which could be related to each com-
ponent in such a way that the sum of such numbers will be related in
the same way to the equally loud reference tone. Such efforts have
failed because the amount contributed by any component toward the
total loudness sensation depends not only upon the properties of this
component but also upon the properties of the other components in
the combination. The answer to the problem of finding a method of
calculating the loudness level lies in determining the nature of the ear
and brain as measuring instruments in evaluating the magnitude of an
auditory sensation.
One can readily estimate roughly the magnitude of an auditory
sensation; for example, one can tell whether the sound is soft or loud.
There have been many theories to account for this change in loudness.
One that seems very reasonable to us is that the loudness experienced
is dependent upon the total number of nerve impulses per second going
to the brain along all the fibers that are excited. Although such an
assumption is not necessary for deriving the formula for calculating
loudness it aids in making the meaning of the quantities involved more
definite.
Let us consider, then, a complex tone having n components each of
which is specified by a value of intensity level 0k and of frequency fk.
Let N be a number which measures the magnitude of the auditory
sensation produced when a typical individual listens to a pure tone.
Since by definition the magnitude of an auditory sensation is the loudness,
then N is the loudness of this simple tone. Loudness as used here must
not be confused with loudness level. The latter is measured by the
intensity of the equally loud reference tone and is expressed in decibels
while the former will be expressed in units related to loudness levels in
a manner to be developed. If we accept the assumption mentioned
above, N is proportional to the number of nerve impulses per second
reaching the brain along all the excited nerve fibers when the typical
observer listens to a simple tone.
Let the dependency of the loudness N upon the frequency / and the
intensity /3 for a simple tone be represented by
N = G(f, /3), (7)
where G is a function which is determined by any pair of values of /
384 BELL SYSTEM TECHNICAL JOURNAL
and /3. For the reference tone, / is 1000 and /3 is equal to the loudness
level L, so a determination of the relation expressed in Eq. (7) for the
reference tone gives the desired relation between loudness and loudness
level.
If now a simple tone is put into combination with other simple tones
to form a complex tone, its loudness contribution, that is, its con-
tribution toward the total sensation, will in general be somewhat less
because of the interference of the other components. For example, if
the other components are much louder and in the same frequency
region the loudness of the simple tone in such a combination will be
zero. Let 1 — b be the fractional reduction in loudness because of its
being in such a combination. Then bN is the contribution of this
component toward the loudness of the complex tone. It will be seen
that b by definition always remains between 0 and unity. It depends
not only upon the frequency and intensity of the simple tone under
discussion but also upon the frequencies and intensities of the other
components. It will be shown later that this dependence can be
determined from experimental measurements.
The subscript k will be used when/ and /3 correspond to the frequency
and intensity level of the &th component of the complex tone, and the
subscript r used when / is 1000 cycles per second. The "loudness
level" L by definition, is the intensity level of the reference tone when
it is adjusted so it and the complex tone sound equally loud. Then
Nr = G(1000, L) = kfbkNk = kfbkG(fk, /3*). (8)
*=i t=i
Now let the reference tone be adjusted so that it sounds equally loud
successively to simple tones corresponding in frequency and intensity
to each component of the complex tone.
Designate the experimental values thus determined as L\, L2, L%, • • •
Lk, ' ' • Ln. Then from the definition of these values
Nk = G(1000, Lk) = G(fk, fik), (9)
since for a single tone bk is unity. On substituting the values from
(9) into (8) there results the fundamental equation for calculating the
loudness of a complex tone
G(1000, L) = X?&*G(1000, Lk). (10)
This transformation looks simple but it is a very important one since
instead of having to determine a different function for every com-
LOUDNESS 385
ponent, we now have to determine a single function depending only
upon the properties of the reference tone and as stated above this
function is the relationship between loudness and loudness level.
And since the frequency is always 1000 this function is dependent only
upon the single variable, the intensity level.
This formula has no practical value unless we can determine bk and
G in terms of quantities which can be obtained by physical measure-
ments. It will be shown that experimental measurements of the
loudness levels L and Lk upon simple and complex tones of a properly
chosen structure have yielded results which have enabled us to find
the dependence of b and G upon the frequencies and intensities of the
components. When b and G are known, then the more general
function G(f, /3) can be obtained from Eq. (9), and the experimental
values of Lk corresponding to ft and /3*.
Determination of the Relation Between Lk, fk and /3fc
This relation can be obtained from experimental measurements of
the loudness levels of pure tones. Such measurements were made by
Kingsbury 4 which covered a range in frequency and intensity limited
by instrumentalities then available. Using the experimental technique
described in Appendix A, we have again obtained the loudness levels
of pure tones, this time covering practically the whole audible range.
(See Appendix B for a comparison with Kingsbury's results.)
All of the data on loudness levels both for pure and also complex
tones taken in our laboratory which are discussed in this paper have
been taken with telephone receivers on the ears. It has been explained
previously how telephone receivers may be used to introduce the
reference tone into the ears at known loudness levels to obtain the
loudness levels of other sounds by a loudness balance. If the receivers
are also used for producing the sounds whose loudness levels are being
determined, then an additional calibration, which will be explained
later,, is necessary if it is desired to know the intensity levels of the
sounds.
The experimental data for determining the relation between Lk and
/* are given in Table I in terms of voltage levels. (Voltage level
= 20 log V, where V is the e.m.f. across the receivers in volts.) The
pairs of values in each double column give the voltage levels of the
reference tone and the pure tone having the frequency indicated at
the top of the column when the two tones coming from the head re-
ceivers were judged to be equally loud when using the technique
4 B. A. Kingsbury, "A Direct Comparison of the Loudness of Pure Tones," Phys.
Rev. 29, 588 (1927).
386
BELL SYSTEM TECHNICAL JOURNAL
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LOUDNESS 387
described in Appendix A. For example, in the second column it will
be seen that for the 125-cycle tone when the voltage is + 9.8 db above
1 volt then the voltage level for the reference tone must be 4.4 db
below 1 volt for equality of loudness. The bottom set of numbers in
each column gives the threshold values for this group of observers.
Each voltage level in Table I is the median of 297 observations
representing the combined results of eleven observers. The method of
obtaining these is explained in Appendix A also. The standard
deviation was computed and it was found to be somewhat larger for
tests in which the tone differed most in frequency from the reference
tone. The probable error of the combined result as computed in the
usual way was between 1 and 2 db. Since deviations of any one
observer's results from his own average are less than the deviations of
his average from the average of the group, it would be necessary to
increase the size of the group if values more representative of the
average normal ear were desired.
The data shown in Table I can be reduced to the number of decibels
above threshold if we accept the values of this crew as the reference
threshold values. However, we have already adopted a value for the
1000-cycle reference zero. As will be shown, our crew obtained a
threshold for the reference tone which is 3 db above the reference
level chosen.
It is not only more convenient but also more reliable to relate the
data to a calibration of the receivers in terms of physical measurements
of the sound intensity rather than to the threshold values. Except in
experimental work where the intensity of the sound can be definitely
controlled, it is obviously impractical to measure directly the threshold
level by using a large group of observers having normal hearing. For
most purposes it is more convenient to measure the intensity levels
0i, /32, • • • 0k, etc., directly rather than have them related in any way
to the threshold of hearing.
In order to reduce the data in Table I to those which one would
obtain if the observers were listening to a free wave and facing the
source, we must obtain a field calibration of the telephone receivers
used in the loudness comparisons. The calibration for the reference
tone frequency has been explained previously and the equation
0T = ft + 10 log (Wr/Wi) (5)
derived for the relation between the intensity /3r of the reference tone
and the electrical power Wr in the receivers. The calibration consisted
of finding by means of loudness balances a power W\ in the receivers
which produces a tone equal in loudness to that of a free wave having
an intensity level fi\.
388 BELL SYSTEM TECHNICAL JOURNAL
For sounds other than the 1000-cycle reference tone a relation
similar to Eq. (5) can be derived, namely,
0 = 01 + 10 log (W/WO, (11)
where 0i and W\ are corresponding values found from loudness balances
for each frequency or complex wave form of interest. If, as is usually
assumed, a linear relation exists between 0 and 10 log W, then de-
terminations of 0i and Wi at one level are sufficient and it follows that
a change in the power level of A decibels will produce a corresponding
change of A decibels in the intensity of the sound generated. Ob-
viously the receivers must not be overloaded or this assumption will
not be valid. Rather than depend upon the existence of a linear
relation between 0 and 10 log W with no confirming data, the receivers
used in this investigation were calibrated at two widely separated
levels.
Referring again to Table I, the data are expressed in terms of voltage
levels instead of power levels. If, as was the case with our receivers,
the electrical impedance is essentially a constant, Eq. (11) can be put
in the form:
0 = 0i + 2Olog(7/F!) (12)
or
0 = 20 log V + C, (13)
where V is the voltage across the receivers and C is a constant of the
receivers to be determined from a calibration giving corresponding
values of 0i and 20 log V\. The calibration will now be described.
By using the sound stage and the technique of measuring field
pressures described by Sivian and White 5 and by using the technique
for making loudness measurements described in Appendix A, the
following measurements were made. An electrical voltage V\ was
placed across the two head receivers such that the loudness level pro-
duced was the same at each frequency. The observer listened to the
tone in these head receivers and then after \\ seconds silence listened
to the tone from the loud speaker producing a free wave of the same
frequency. The voltage level across the loud speaker necessary to
produce a tone equally loud to the tone from the head receivers was
obtained using the procedure described in Appendix A. The free wave
intensity level 0i corresponding to this voltage level was measured in
the manner described in Sivian and White's paper. Threshold values
both for the head receivers and the loud speaker were also observed.
In these tests eleven observers were used. The results obtained are
given in Table II. In the second row values of 20 log Vu the voltage
? L. J. Sivian and S. D. White, "Minimum Audible Sound Fields," Jour. Acous.
Soc- Am. 4, 288 (1933).
LOUDNESS
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390
BELL SYSTEM TECHNICAL JOURNAL
level, are given. The intensity levels, /3i, of the free wave which
sounded equally loud are given in the third row. In the fourth row
the values of the constant C, the calibration we are seeking, are given.
The voltage level added to this constant gives the equivalent free
wave intensity level. In the fifth, sixth and seventh rows, similar
values are given which were determined at the threshold level. In the
bottom row the differences in the constants determined at the two
levels are given. The fact that the difference is no larger than the
probable error is very significant. It means that throughout this
wide range there is a linear relationship between the equivalent field
intensity levels, /3, and the voltage levels, 20 log V, so that the
formula (13)
0 = 20 log V + C
can be applied to our receivers with considerable confidence.
£ 60
i_
t's
/
\
„..^
\
\
\
\
500 1000
FREQUENCY IN CYCLES PER SECOND
5000 10000 20000
Fig. 1 — Field calibration of loudness balance receivers.6 (Calibration made at
I = 60 db.)
The constant C determined at the high level was determined with
greater accuracy than at the threshold. For this reason only the
values for the higher level were used for the calibration curve. Also
in these tests only four receivers were used while in the loudness tests
eight receivers were used. The difference between the efficiency of
the former four and the latter eight receivers was determined by
measurements on an artificial ear. The figures given in Table II
were corrected by this difference. The resulting calibration curve is
that given in Fig. 1. It should be pointed out here that such a calibra-
tion curve on a single individual would show considerable deviations
from this average curve. These deviations are real, that is, they are
due to the sizes and shapes of the ear canals.
6 The ordinates represent the intensity level in db of a free wave in air which,
when listened to with both ears in the standard manner, is as loud as a tone of the
same frequency heard from the two head receivers used in the tests when an e.m.f. of
one volt is applied to the receiver terminals.
LOUDNESS
391
We can now express the data in Table I in terms of field intensity
levels. To do this, the data in each double column were plotted and a
smooth curve drawn through the observed points. The resulting
curves give the relation between voltage levels of the pure tones for
equality of loudness. From the calibration curve of the receivers
these levels are converted to intensity levels by a simple shift in the
axes of coordinates. Since the intensity level of the reference tone is
by definition the "loudness level," these shifted curves wi'l represent
the loudness level of pure tones in terms of intensity levels. The
resulting curves for the ten tones tested are given in Figs. 2A to 2J.
Each point on these curves corresponds to a pair of values in Table I
except for the threshold values. The results of separate determina-
tions by the crew used in these loudness tests at different times are
given by the circles. The points represented by (*) are the values
adopted by Sivian and White. It will be seen that most of the
80
J 20
/o
/ 6
20» TO
NE
*£0
p
/
12
i'Xj TO
NE
V
/
60
40
20
q/**
2500.
TONE
fe
si
/
s>ooa
TONE
•
/
20 40
60
C
100 120 0 20 40
INTENSITY LEVEL-DB
60
D
Fig. 2 (A to D) — Loudness levels of pure tones.
392
BELL SYSTEM TECHNICAL JOURNAL
120
100
80
60
40
20
m
Z 120
Q
3 100
SO
60
40
20
0
120
100
ID
Q
_;, 80
ul
>
bJ
-i 60
ui
Z 40
Q
3 20
0
°cS
200
D~ TONE
•
4000~ TONE
/
/
565C
~ TO
NE
I
/
/
'
■$/
eoo
D~ TONE
y
/
/
v
0 20 40 60 80 100 120
0 20 40 60 80 100 120
INTENSITY LEVEL-DB
H
%
n °S
II30C
~ TC
NE
c
1600
0~ T
DNE
0 20 40 60 80 100 120 0 20 40 60 80 100 120
I INTENSITY LEVEL-DB j
Fig. 2 (E to J) — Loudness levels of pure tones.
LOUDNESS
393
threshold points are slightly above the zero we have chosen. This
means that our zero corresponds to the thresholds of observers who
are slightly more acute than the average.
From these curves the loudness level contours can be drawn. The
first set of loudness level contours are plotted with levels above
reference threshold as ordinates. For example, the zero loudness level
contour corresponds to points where the curves of Figs. 2A to 2J
intersect the abscissa axis. The number of db above these points is
plotted as the ordinate in the loudness level contours shown in Fig. 3.
From a consideration of the nature of the hearing mechanism we
believe that these curves should be smooth. These curves, therefore,
100 500 1000 5000 10000 20000
FREQUENCY IN CYCLES PER SECOND
Fig. 3 — Loudness level contours.
represent the best set of smooth curves which we could draw through
the observed points. After the smoothing process, the curves in
Figs. 2 A to 2 J were then adjusted to correspond. The curves shown
in these figures are such adjusted curves.
In Fig. 4 a similar set of loudness level contours is shown using
intensity levels as ordinates. There are good reasons 5 for believing
that the peculiar shape of these contours for frequencies above 1000
c.p.s. is due to diffraction around the head of the observer as he faces
the source of sound. It was for this reason that the smoothing process
was done with the curves plotted with the level above threshold as
the ordinate.
6 Loc. cit.
394
BELL SYSTEM TECHNICAL JOURNAL
From these loudness level contours, the curves shown in Figs. 5A
and 5B were obtained. They show the loudness level vs. intensity
level with frequency as a parameter. They are convenient to use for
calculation purposes.
It is interesting to note that through a large part of the practical
range for tones of frequencies from 300 c.p.s. to 4000 c.p.s. the loudness
level is approximately equal to the intensity level. From these curves,
it is possible to obtain any value of Z,* in terms of 0k and /*.
100 500 1000
FREQUENCY IN CYCLES PER SECOND
Fig. 4. — Loudness level contours.
5000 100OO
On Fig. 4 the 120-db loudness level contour has been marked
"Feeling." The data published by R. R. Riesz 7 on the threshold of
feeling indicate that this contour is very close to the feeling point
throughout the frequency range where data have been taken.
Determination of the Loudness Function G
In the section " Formulation of the Empirical Theory for Calculating
the Loudness of a Steady Complex Tone," the fundamental equation
for calculating the loudness level of a complex tone was derived,
7 R. R. Riesz, "The Relationship Between Loudness and the Minimum Perceptible
Increment of Intensity," Jour. Aeons. Soc. Am. 4, 211 (1933).
LOUDNESS
395
120
-10
-20 -10
10 20 30 40 50 60 70 80 90 100 110 120
INTENSITY LEVEL-DB
100
90
80
60
50
40
30
10
■
/
/
/
y/
$y
r$
u
Y
i
f
/
" *
n
/
A
x;
/
/
-"20 -10 0 10 20 30 40 50 60 70 80 90 100 110 120
INTENSITY LEVEL-DB
Fig. 5 (A and B) — Loudness levels of pure tones.
396 BELL SYSTEM TECHNICAL JOURNAL
namely,
G(1000, L) = 5 &*G(1000, Lk). (10)
i=l
If the type of complex tone can be chosen so that bk is unity and also
so that the values of Lk for each component are equal, then the funda-
mental equation for calculating loudness becomes
' G{L) = nG(Lk), (14)
where n is the number of components. Since we are always dealing
in this section with £(1000, L) or G(1000, Lk), the 1000 is left out in
the above nomenclature. If experimental measurements of L corre-
sponding to values of Lk are taken for a tone fulfilling the above con-
ditions throughout the audible range, the function G can be determined.
If we accept the theory that, when two simple tones widely separated
in frequency act upon the ear, the nerve terminals stimulated by each
are at different portions of the basilar membrane, then we would
expect the interference of the loudness of one upon that of the other
would be negligible. Consequently, for such a combination b is unity.
Measurements were made upon two such tones, the two components
being equally loud, the first having frequencies of 1000 and 2000
cycles and the second, frequencies of 125 and 1000 cycles. The ob-
served points are shown along the second curve from the top of Fig. 6.
The abscissae give the loudness level Lk of each component and the
ordinates the loudness level L of the two components combined. The
equation G(y) = 2G(x) should represent these data. Similar measure-
ments were made with a complex tone having 10 components, all
equally loud. The method of generating such tones is described in
Appendix C. The results are shown by the points along the top
curve of Fig. 6. The equation G(y) — 10G(x) should represent these
data except at high levels where bk is not unity.
There is probably a complete separation between stimulated patches
of nerve endings when the first component is introduced into one ear
and the second component into the other ear. In this case the same
or different frequencies can be used. Since it is easier to make loud-
ness balances when the same kind of sound is used, measurements were
made (1) with 125-cycle tones (2) with 1000-cycle tones and (3) with
4000-cycle tones. The results are shown on Fig. 7. In this curve the
ordinates give the loudness levels when one ear is used while the
abscissae give the corresponding loudness levels for the same intensity
level of the tone when both ears are used for listening. If binaural
versus monaural loudness data actually fit into this scheme of calcula-
LOUDNESS
397
O 80
Ul
z
m
2
O
40
bK=1
/ /
•
•
4
/
./
•-TEN TONES Af=530^
TWO TONES
o-IOOO AND 2000 <v
®-125 AND 1000 -\j
i
/
/,
"0 20 40 60 80 100 120 140
LOUDNESS LEVEL OF EACH COMPONENT- DB
Fig. 6— Complex tones having components widely separated in frequency.
120
a. 80
/a
/
/
/
/
/
a- 1000 -v.
A-125^
x-4000'V-
s
S
/
/
y
*
/
s
40 60 80 100
LOUDNESS LEVEL. BOTH EARS-DB
Fig. 7 — Relation between loudness levels listening with one ear and with both ears.
398 BELL SYSTEM TECHNICAL JOURNAL
tion these points should be represented by
G(y) = $G(x).
Any one of these curves which was accurately determined would be
sufficient to completely determine the function G.
For example, consider the curve for two tones. It is evident that
it is only necessary to deal with relative values of G so that we can
choose one value arbitrarily. The value of G(0) was chosen equal to
unity. Therefore,
G(0) = 1,
G(y0) = 2G(0) = 2 where yo corresponds to x = 0,
G(yi) = 2G(xi) = 2G(yo) — 4 where y\ corresponds to X\ = yo,
G(y2) = 2G(x2) — 2G(yi) = 8 where y2 corresponds to x2 = y\,
G(yk) = 2G(xk) = 2G{yk-\) — 2k+l where y* corresponds to xh = yk-i.
In this way a set of values for G can be obtained. A smooth curve
connecting all such calculated points will enable one to find any value
of G(x) for a given value of x. In a similar way sets of values can be
obtained from the other two experimental curves. Instead of using
any one of the curves alone the values of G were chosen to best fit all
three sets of data, taking into account the fact that the observed
points for the 10-tone data might be low at the higher levels where b
would be less than unity. The values for the function which were
finally adopted are given in Table III. From these values the three
solid curves of Figs. 6 and 7 were calculated by the equations
G{y) = 10G(x), G(y) = 2G{x), G(y) = \G{x).
The fit of the three sets of data is sufficiently good, we think, to justify
the point of view taken in developing the formula. The calculated
points for the 10-component tones agree with the observed ones when
the proper value of bk is introduced into the formula. In this con-
nection it is important to emphasize that in calculating the loudness
level of a complex tone under the condition of listening with one ear
instead of two, a factor of \ must be placed in front of the summation
of Eq. (10). This will be explained in greater detail later. The values
of G for negative values of L were chosen after considering all the data
on the threshold values of the complex tones studied. These data will
be given with the other loudness data on complex tones. It is in-
teresting to note here that the threshold data show that 10 pure tones,
which are below the threshold when sounded separately, will combine
LOUDNESS
399
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400 BELL SYSTEM TECHNICAL JOURNAL
to give a tone which can be heard. When the components are all in
the high pitch range and all equally loud, each component may be
from 6 to 8 db below the threshold and the combination will still be
audible. When they are all in the low pitch range they may be only
2 or 3 db below the threshold. The closeness of packing of the com-
ponents also influences the threshold. For example, if the ten
components are all within a 100-cycle band each one may be down
10 db. It will be shown that the formula proposed above can be made
to take care of these variations in the threshold.
There is still another method which might be used for determining
this loudness function G(L), provided one's judgment as to the magni-
tude of an auditory sensation can be relied upon. If a person were
asked to judge when the loudness of a sound was reduced to one half
it might be expected that he would base his judgment on the experience
of the decrease in loudness when going from the condition of listening
with both ears to that of listening with one ear. Or, if the magnitude
of the sensation is the number of nerve discharges reaching the brain
per second, then when this has decreased to one half, he might be able
to say that the loudness has decreased one half.
In any case, if it is assumed that an observer can judge when the
magnitude of the auditory sensation, that is, the loudness, is reduced to
one half, then the value of the loudness function G can be computed
from such measurements.
Several different research workers have made such measurements.
The measurements are somewhat in conflict at the present time so
that they did not in any way influence the choice of the loudness
function. Rather we used the loudness function given in Table III
to calculate what such observations should give. A comparison of
the calculated and observed results is given below. In Table IV is
shown a comparison of calculated and observed results of data taken
by Ham and Parkinson.8 The observed values were taken from
Tables la, lb, 2a, 2b, 3a and 3b of their paper. The calculation is
very simple. From the number of decibels above threshold 5 the
loudness level L is determined from the curves of Fig. 3. The frac-
tional reduction is just the fractional reduction in the loudness function
for the corresponding values of L. The agreement between observed
and calculated results is remarkably good. However, the agreement
with the data of Laird, Taylor and Wille is very poor, as is shown by
Table V. The calculation was made only for the 1024-cycle tone.
The observed data were taken from Table VII of the paper by Laird,
8 L. B. Ham and J. S. Parkinson, "Loudness and Intensity Relations," Jour. Acous.
Soc. Am. 3, 511 (1932).
TABLE IV
Comparison of Calculated and Observed Fractional Loudness (Ham and
Parkinson)
350 Cycles
Fractional Reduction in Loudness
Cal. %
Obs. %
74.0
70.4
67.7
64.0
59.0
54.0
44.0
34.0
85
82
79
75
70
65
53
41
25,000
19,800
15,800
11,400
7,950
5,870
2,680
1,100
100
79
63
46
32
24
11
4
100.0
83.0
67.0
49.0
35.0
26.0
15.0
8.0
59.5
57.7
55.0
49.0
44.0
39.0
34.0
24.0
71
69
66
59
53
47
41
29
8,510
7,440
6,240
4,070
2,680
1,780
1,060
324
100
87
73
48
31
21
12
4
100.0
92.0
77.0
57.0
38.0
25.0
13.0
6.0
1000 Cycles
86.0
86
27,200
100
100.0
82.4
82
19,800
73
68.0
79.7
80
17,100
63
53.0
76.0
76
12,400
46
41.0
71.0
71
8,510
31
26.0
66.0
66
6,420
24
20.0
56.0
56
3,310
12
13.0
46.0
46
1,640
6
8.0
56.0
56
3,310
100
100.0
54.2
54
2,880
87
93.4
51.5
52
2,510
76
74.6
48.8
49
2,070
62
55.0
46.0
46
1,640
49
40.9
41.0
41
1,060
32
24.5
36.0
36
675
20
10.8
2500 Cycles
74.0
69
7,440
100
100.0
70.4
64
5,560
75
86.4
67.7
62
4,950
67
68.1
64.0
58
3,820
51
49.5
59.0
53
2,680
36
32.8
54.0
48
1,920
26
23.3
44.0
39
890
12
13.0
34.0
30
360
5
6.7
44.0
39
890
100
100.0
42.2
37
740
83
94.6
39.5
36
675
76
82.2
36.8
33
505
57
61.1
34.0
30
360
41
46.0
29.0
26
222
25
27.8
24.0
21
113
13
14.9
402
BELL SYSTEM TECHNICAL JOURNAL
TABLE V
Comparison of Calculated and Observed Fractional Loudness (Laird, Taylor
and Wille)
Level for J Loudness Reduction
Original Loudness
Cal. Level for i
Level
Cal.
Obs.
100
92
76.0
84
90
82
68.0
73
80
71
60.0
60
70
58
49.5
48
60
50
40.5
41
50
42
31.0
34
40
33
21.0
27
30
25
14.9
20
20
16
6.5
13
10
7
5.0
4
Taylor and Wille.9 As shown in Table V the calculation of the level
for one fourth reduction in loudness agrees better with the observed
data corresponding to one half reduction in loudness.
Firestone and Geiger reported some preliminary values which Were
in closer agreement with those obtained by Parkinson and Ham, but
their completed paper has not yet been published.10 Because of the
lack of agreement of observed data of this sort we concluded that it
could not be used for influencing the choice of the values of the loud-
ness function adopted and shown in Table III. It is to be hoped that
more data of this type will be taken until there is a better agreement
between observed results of different observers. It should be empha-
sized here that changes of the level above threshold corresponding to
any fixed increase or decrease in loudness will, according to the theory
outlined in this paper, depend upon the frequency of the tone when
using pure tones, or upon its structure when using complex tones.
Determination of the Formula for Calculating bk
Having now determined the function G for all values of L or Lk we
can proceed to find methods of calculating bk- Its value is evidently
dependent upon the frequency and intensity of all the other com-
ponents present as well as upon the component being considered.
For practical computations, simplifying assumptions can be made. In
most cases the reduction of bk from unity is principally due to the
adjacent component on the side of the lower pitch. This is due to the
fact that a tone masks another tone of higher pitch very much more
9 Laird, Taylor and Wille, "The Apparent Reduction in Loudness," Jour. Acous.
Soc. Am. 3, 393 (1932).
10 This paper is now available. P. H. Geiger and F. A. Firestone, "The Estimation
of Fractional Loudness," Jour. Acous. Soc. Am. 5, 25 (1933).
LOUDNESS 403
than one of lower pitch. For example, in most cases a tone which is
100 cycles higher than the masking tone would be masked when it is
reduced 25 db below the level of the masking tone, whereas a tone 100
cycles lower in frequency will be masked only when it is reduced from
40 to 60 db below the level of the masking tone. It will therefore be
assumed that the neighboring component on the side of lower pitch
which causes the greatest masking will account for all the reduction
in bk- Designating this component with the subscript m, meaning
the masking component, then we have bt expressed as a function of
the following variables.
h = B(fk,fm,Sh,Sm), (15)
where / is the frequency and 5 is the level above threshold. For the
case when the level of the &th component is T db below the level of
the masking component, where T is just sufficient for the component to
be masked, then the value of b would be equal to zero. Also, it is
reasonable to assume that when the masking component is at a level
somewhat less than T db below the &th component, the latter will
have a value of 6* which is unity. It is thus seen that the fundamental
of a series of tones will always have a value of &* equal to unity.
For the case when the masking component and the £th component
have the same loudness, the function representing bk will be con-
siderably simplified, particularly if it were also found to be independent
of fk and only dependent upon the difference between /* and/m. From
the theory of hearing one would expect that this would be approxi-
mately true for the following reasons:
The distance in millimeters between the positions of maximum
response on the basilar membrane for the two components is more
nearly proportional to differences in pitch than to differences in fre-
quency. However, the peaks are sharpest in the high frequency
regions where the distances on the basilar membrane for a given A/
are smallest. Also, in the low frequency region where the distances
for a given A/ are largest, these peaks are broadest. These two
factors tend to make the interference between two components having
a fixed difference in frequency approximately the same regardless of
their position on the frequency scale. However, it would be extra-
ordinary if these two factors just balanced. To test this point three
complex tones having ten components with a common A/ of 50 cycles
were tested for loudness. The first had frequencies of 50-100-150- • •
500, the second 1400-1450- • -1900, and the third 3400-3450- • -3900.
The results of these tests are shown in Fig. 8. The abscissae give the
loudness level of each component and the ordinates the measured loud-
404
BELL SYSTEM TECHNICAL JOURNAL
ness level of the combined tone. Similar results were obtained with a
complex tone having ten components of equal loudness and a common
frequency difference of 100 cycles. The results are shown in Fig. 9.
It will be seen that although the points corresponding to the different
120
110
100
90
60
> 50
a 40
20
10
-10
O- F
UNDAfc
i
i
ENTAL
- 50«j
= 1400
= 3400
/ ,
A-
^
/i
//
/oj
/
'f
J9l/
*/(/
V
•/'
a/
U
a
•
//
' 1
-20
10 20 30 40 50 60 70 80
LOUDNESS LEVEL OF EACH COMPONENT-DB
100
Fig. 8 — Loudness levels of complex tones having ten equally loud components 50
cycles apart.
frequency ranges lie approximately upon the same curve through the
middle range, there are consistent departures at both the high and
low intensities. If we choose the frequency of the components largely
in the middle range then this factor b will be dependent only upon A/
and Lfc.
LOUDNESS
405
To determine the value of b for this range in terms of A/ and Lk, a
series of loudness measurements was made upon complex tones having
ten components with a common difference in frequency A/ and all
having a common loudness level Lk- The values of A/ were 340, 230,
no
100
90
z
o
I 70
o
u
o
z*°
o
-I
u
> 50
w 40
o
D
o
■J 30
o-F
■ •-
a -
JNDAM
II
:ntal=
100 a.
1000 a-
3000 a.
/* *
//
'yr
/A
/
r/
/
I
/
V /
/
/
p/fey
/
/
r
//
10 20 30 40 50 60 70 80
LOUDNESS LEVEL OF EACH COMPONENT-DB
Fig. 9— Loudness levels of complex tones having ten equally loud components 100
cycles apart.
112 and 56 cycles per second. The fundamental for each tone was
close to 1000 cycles. The ten-component tones having frequencies
which are multiples of 530 was included in this series. The results of
loudness balances are shown by the points in Fig. 10.
By taking all the data as a whole, the curves were considered to
406
BELL SYSTEM TECHNICAL JOURNAL
give the best fit. The values of b were calculated from these curves as
follows:
According to the assumptions made above, the component of lowest
pitch in the series of components always has a value of bk equal to
unity. Therefore for the series of 10 components having a common
100
90
IE
S 80
<n
Z 70
o
0-
o
O 60
50
' X
/x >•
y /
/m
/
4
y
o FREQUENCY DIFFERENCE" 340
b. " " =230
x " " =112
• " " = 50
/
/
•ax
/
f
10
-10 ° 0
10 20 30 40 50 60 70
LOUDNESS LEVEL OF SINGLE COMPONENT-DB
80
90
Fig. 10 — Loudness levels of complex tones having ten equally loud components with
a fundamental frequency of 1000 c.p.s.
loudness level Lk, the value of L is related to Lk by
G(L) = (1 + 9bk)G(Lk)
or by solving for bk
bk = (l/9)[G(L)/G(Lk) - 1].
(16)
The values of bk can be computed from this equation from the ob-
served values of L and Lk by using the values of G given in Table III.
Because of the difficulty in obtaining accurate values of L and Lk such
computed values of bk will be rather inaccurate. Consequently, con-
siderable freedom is left in choosing a simple formula which will
LOUDNESS
407
represent the results. When the values of bk derived in this way were
plotted with bk as ordinates and A/ as abscissae and Lk as a variable
parameter then the resulting graphs were a series of straight lines
going through the common point (— 250, 0) but having slopes de-
pending upon Lk. Consequently the following formula
bk = [(250 + A/)/1000]<2(L*)
(17)
will represent the results. The quantity A/ is the common difference
in frequency between the components, Lk the loudness level of each
component, and Q a function depending upon Lk. The results indi-
cated that Q could be represented by the curve in Fig. 11.
\
4.0
\
^
\
O 3.0
o
\
\
Id
_l
£ 2.0
s
1.0
20
40 60 80 100
LOUDNESS LEVEL OF COMPONENT-DB
120
Fig. 1 1 — Loudness factor Q.
Also the condition must be imposed upon this equation that b is
always taken as unity whenever the calculation gives values greater
than unity. The solid curves shown in Fig. 10 are actually calculated
curves using these equations, so the comparison of these curves with
the observed points gives an indication of how well this equation fits
the data. For this series of tones Q could be made to depend upon
/3a- rather than Lk and approximately the same results would be ob-
tained since /3a- and Lk are nearly equal in this range of frequencies.
However, for tones having low intensities and low frequencies, pk will
be much larger than Lk and consequently Q will be smaller and hence
the calculated loudness smaller. The results in Figs. 8 and 9 are just
408
BELL SYSTEM TECHNICAL JOURNAL
contrary to this. To make the calculated and observed results agree
with these two sets of data, Q was made to depend upon
x = ^ + 30 log/- 95
instead of Lk.
It was found when using this function of 0 and / as an abscissa and
the same ordinates as in Fig. 10, a value of Q was obtained which gives
just as good a fit for the data of Fig. 10 and also gives a better fit for
the data of Figs. 8 and 9. Other much more complicated factors were
tried to make the observed and calculated results shown in these two
figures come into better agreement but none were more satisfactory
than the simple procedure outlined above. For purpose of calculation
the values of Q are tabulated in Table VI.
TABLE VI
Values of Q(X)
X
0
l
2
3
4
5
6
7
8
9
0
5.00
4.88
4.76
4.64
4.53
4.41
4.29
4.17
4.05
3.94
10
3.82
3.70
3.58
3.46
3.35
3.33
3.11
2.99
2.87
2.76
20
2.64
2.52
2.40
2.28
2.16
2.05
1.95
1.85
1.76
1.68
30
1.60
1.53
1.47
1.40
1.35
1.30
1.25
1.20
1.16
1.13
40
1.09
1.06
1.03
1.01
0.99
0.97
0.95
0.94
0.92
0.91
50
0.90
0.90
0.89
0.89
0.88
0.88
0.88
0.88
0.88
0.88
60
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.89
0.89
0.90
70
0.90
0.91
0.92
0.93
0.94
0.96
0.97
0.99
1.00
1.02
80
1.04
1.06
1.08
1.10
1.13
1.15
1.17
1.19
1.22
1.24
90
1.27
1.29
1.31
1.34
1.36
1.39
1.41
1.44
1.46
1.48
100
1.51
1.53
1.55
1.58
1.60
1.62
1.64
1.67
1.69
1.71
Note: X = /3* + 30 log/* - 95.
There are reasons based upon the mechanics of hearing for treating
components which are very close together by a separate method.
When they are close together the combination must act as though the
energy were all in a single component, since the components act upon
approximately the same set of nerve terminals. For this reason it
seems logical to combine them by the energy law and treat the com-
bination as a single frequency. That some such procedure is necessary
is shown from the absurdities into which one is led when one tries to
make Eq. (17) applicable to all cases. For example, if 100 components
were crowded into a 1000-cycle space about a 1000-cycle tone, then it
is obvious that the combination should sound about 20 db louder.
But according to Eq. (10) to make this true for values of Lk greater
than 45, bk must be chosen as 0.036. Similarly, for 10 tones thus
crowded together L — Lk must be about 10 db and therefore bk = 0.13
and then for two such tones L — Lk must be 3 db and the corresponding
LOUDNESS 409
value of bk = 0.26. These three values must belong to the same
condition A/ = 10. It is evident then that the formulae for b given by
Eq. (17) will lead to very erroneous results for such components.
In order to cover such cases it was necessary to group together all
components within a certain frequency band and treat them as a single
component. Since there was no definite criterion for determining
accurately what these limiting bands should be, several were tried and
ones selected which gave the best agreement between computed and
observed results. The following band widths were finally chosen :
For frequencies below 2000 cycles, the band width is 100 cycles; for
frequencies between 2000 and 4000 cycles, the band width is 200
cycles; for frequencies between 4000 and 8000 cycles, the band width
is 400 cycles; and for frequencies between 8000 and 16,000 cycles, the
band width is 800 cycles. If there are k components within one of
these limiting bands, the intensity / taken for the equivalent single
frequency component is given by
I = £ h = £ 10"*/10. (18)
A frequency must be assigned to the combination. It seems reasonable
to assign a weighted value of / given by the equation
/ = Ehh/I = IMOW'VI io<"10. (19)
Only a small error will be introduced if the mid-frequency of such
bands be taken as the frequency of an equivalent component except for
the band of lowest frequency. Below 125 cycles it is important that
the frequency and intensity of each component be known, since in
this region the loudness level Lk changes very rapidly with both changes
in intensity and frequency. However, if the intensity for this band
is lower than that for other bands, it will contribute little to the total
loudness so that only a small error will be introduced by a wrong choice
of frequency for the band.
This then gives a method of calculating bk when the adjacent com-
ponents are equal in loudness. When they are not equal let us define
the difference AL by
AL = Lk - Lm. (20)
Also let this difference be T when Lm is adjusted so that the masking
component just masks the component k. Then the function for
calculating b must satisfy the following conditions:
bk = [(250 + A/)/1000]<2 when AL = 0,
bk = 0 when AL = - T.
410
BELL SYSTEM TECHNICAL JOURNAL
Also the following condition when Lk is larger than Lm must be satisfied,
namely, bk = 1 when AL = some value somewhat smaller than + T.
The value of T can be obtained from masking curves. An examination
of these data indicates that to a good approximation the value of T is
dependent upon the single variable /* — 2fm. A curve showing the
relation between T and this variable is shown in Fig. 12. It will be
seen that for most practical cases the value of T is 25. It cannot be
claimed that the curve of Fig. 12 is an accurate representation of the
masking data, but it is sufficiently accurate for the purpose of loudness
calculation since rather large changes in T will produce a very slight
change in the final calculated loudness level.
80
60
h
a 4°
5
20
-loo
fm= MASKING TONE,
f- MASKED TONE ,
FOR CASE WHEN f >fm
-100 0 100 300 500
VALUES OF Af-fm=f -2fm
Fig;. 12 — Values of the masking: T.
1100
Data were taken in an effort to determine how this function de-
pended upon AL but it was not possible to obtain sufficient accuracy
in the experimental results. The difference between the resultant
loudness level when half the tones are down so as not to contribute to
loudness and when these are equal is not more than 4 or 5 db, which is
not much more than the observational errors in such results.
A series of tests were made with tones similar to those used to obtain
the results shown in Figs. 8 and 9 except that every other component
was down in loudness level 5 db. Also a second series was made in
which every other component was down 10 db. Although these data
were not used in determining the function described above, it was
useful as a check on the final equations derived for calculating the
loudness of tones of this sort.
The factor finally chosen for representing the dependence of bk upon
AL is 10AL/r. This factor is unity for AL = 0, fulfilling the first
LOUDNESS 411
condition mentioned above. It is 0.10 instead of zero for AL = — 25,
the most probable value of T. For A/ = 100 and Q = 0.88 we will
obtain the smallest value of bk without applying the AL factor, namely,
0.31. Then when using this factor as given above, all values of bk will
be unity for values of AL greater than 12 db.
Several more complicated functions of AL were tried but none of
them gave results showing a better agreement with the experimental
values than the function chosen above.
The formula for calculation of bk then becomes
bk = [(250 + fk - fm)/lQ00]Ml"rz"*»TQ(Pk + 30 log/* - 95) (21)
where
fk is the frequency of the component expressed in cycles per second,
fm is the frequency of the masking component expressed in cycles per
second,
Lk is the loudness level of the kt\\ component when sounding alone,
Lm is the loudness level of the masking tone,
Q is a function depending upon the intensity level /3* and the fre-
quency/* of each component and is given in Table VI as a function
of x = ft + 30 log/* - 95,
T is the masking and is given by the curve of Fig. 12.
It is important to remember that bk can never be greater than unity so
that all calculated values greater than this must be replaced with values
equal to unity. Also all components within the limiting frequency
bands must be grouped together as indicated above. It is very helpful to
remember that any component for which the loudness level is 12 db
below the jfeth component, that is, the one for which b is being calcu-
lated, need not be considered as possibly being the masking com-
ponent. If all the components preceding the feth are in this class then
bk is unity.
Recapitulation
With these limitations the formula for calculating the loudness level
L of a steady complex tone having n components is
k=n
G(L) = L bkG{Lk), (10)
k=l
where bk is given by Eq. (21). If the values of/* and /3* are measured
directly then corresponding values of L* can be found from Fig. 5.
412
BELL SYSTEM TECHNICAL JOURNAL
Having these values, the masking component can be found either by
inspection or better by trial in Eq. (21). That component whose
values of Lm, fm and T introduced into this equation gives the smallest
value of bk is the masking component.
The values of G and Q can be found from Tables III and VI from
the corresponding values of Lk, @k, and /*. If all these values are now
introduced into Eq. (10), the resulting value of the summation is the
loudness of the complex tone. The loudness level L corresponding to
it is found from Table III.
If it is desired to know the loudness obtained if the typical listener
used only one ear, the result will be obtained if the summation indicated
in Eq. (10) is divided by 2. Practically the same result will be ob-
tained in most instances if the loudness level Lk for each component
when listened to with one ear instead of both ears is inserted in Eq. (10).
(G(Lk) for one ear listening is equal to one half G{Lk) for listening with
both ears for the same value of the intensity level of the component.)
If two complex tones are listened to, one in one ear and one in the
other, it would be expected that the combined loudness would be
the sum of the two loudness values calculated for each ear as though
no sound were in the opposite ear, although this has not been confirmed
by experimental trial. In fact, the loudness reduction factor bk has
been derived from data taken with both ears only, so strictly speaking,
its use is limited to this type of listening.
To illustrate the method of using the formula the loudness of two
complex tones will be calculated. The first may represent the hum
from a dynamo. Its components are given in the table of com-
putations.
Computations
*
fk
A
Lk
Gk
bk
1
60
50
3
3
1.0
2
180
45
25
197
1.0
SftjtGt = 1009
3
300
40
30
360
1.0
4
540
30
27
252
1.0
L = 40
5
1200
25
25
197
1.0
The first step is to find from Fig. 5 the values of Lk from /* and /3*.
Then the loudness values Gt, are found from Table III. Since the
values of L are low and the frequency separation fairly large, one
familiar with these functions would readily see that the values of b
would be unity and a computation would verify it so that the sum of
the G values gives the total loudness 1009. This corresponds to a
loudness level of 40.
LOUDNESS
The second tone calculated is this same hum amplified 30 db.
better illustrates the use of the formula.
Computations
413
It
k
/*
fa
Lk
Gk
U
Un
(30 log h -95)
Q
b
b XG
1
60
80
69
7440
—
—
1.00
7440
7
180
75
n
9130
60
69
-28
0.91
0.41
3740
3
300
70
69
7440
180
72
-21
0.91
0.27
2010
4
540
60
60
4350
300
69
-13
0.94
0.23
1000
5
1200
55
55
3080
540
60
- 3
0.89
0.61
1880
loudness G
= 16070
loudness level L
= 79 db
The loudness level of the combined tones is only 7 db above the
loudness level of the second component. If only one ear is used in
listening, the loudness of this tone is one half, corresponding to a
loudness level of 70 db.
Comparison of Observed and Calculated Results on the Loud-
ness Levels of Complex Tones
In order to show the agreement between observed loudness levels
and levels calculated by means of the formula developed in the pre-
ceding sections, the results of a large number of tests are given here,
including those from which the formula was derived. In Tables VII
to XIII, the first column shows the frequency range over which the
components of the tones were distributed, the figures being the fre-
quencies of the first and last components. Several tones having two
components were tested, but as the tables indicate, the majority of
the tones had ten components. Because of a misunderstanding in the
TABLE VII
Two Component Tones (AL = 0)
Frequency Range
A/
Loudness Levels (db)
u
83
63
43
23
2
1000-1100
100
87
68
47
28
2
Lcalc.
87
68
47
28
4
u
83
63
43
L\
-1
1000-2000
1000
89
71
49
28
2
Lealc.
91
74
52
^28
1
U
84
125-1000
875
Lobs.
Lcalc.
92
92
414
BELL SYSTEM TECHNICAL JOURNAL
TABLE VIII
Ten Component Tones (AL = 0)
Frequency Range
4/
Loudness
Levels (db)
Lk
67
54
33
21
11
-1
50-500
50
83
68
47
38
20
2
£calc.
81
72
53
39
24
8
Lk
78
61
41
23
13
-1
50-500
50
92
73
53
42
25
2
•Lcalc.
91
77
60
42
27
8
Lk
78
69
50
16
6
-1
1400-1895
55
94
82
62
32
22
2
■Lcalc.
93
83
65
31
17
0
Lk
57
37
20
3
1400-1895
55
■Lobs.
Lcalc.
68
73
50
52
34
36
2
5
Lk
84
64
43
24
2
84
64
43
24
2
100-1000
100
95
83
59
41
2
94
80
63
44
2
■^cnlc.
100
83
68
47
12
100
83
68
47
12
Lk
81
64
43
23
13
-4
100-1000
100
93
82
65
49
33
2
•Lcalc.
98
83
68
45
27
3
Lk
83
63
43
23
0
100-1000
100
L>obs.
-Lcalc.
95
99
79
82
59
68
43
45
2
9
Lk
83
63
43
23
78
59
48
27
-7
3100-3900
100
L0hB.
100
82
59
32
99
81
62
38
2
Lcalc.
100
80
60
38
95
77
65
42
0
Lk
79
60
41
17
7
-4
1100-3170
230
100
81
65
33
22
2
Lcalc.
100
83
64
34
18
3
Lk
79
62
42
23
13
-2
260-2600
260
97
82
65
44
28
2
•^•calc.
100
85
68
45
27
5
Lk
75
53
43
25
82
61
43
17
-2
530-5300
530
LobB.
100
83
73
52
105
90
73
40
2
Lcalc.
101
82
72
48
108
89
72
34
5
Lk
61
41
21
-3
530-5300
530
Lobs.
LCalc.
89
89
69
70
45
42
2
4
design of the apparatus for generating the latter tones, a number of
them contained eleven components, so for purposes of identification,
these are placed in a separate group. In the second column of the
tables, next to the frequency range of the tones, the frequency differ-
ence (A/) between adjacent components is given. The remainder of
LOUDNESS
415
TABLE IX
Eleven Component Tones (AL = 0)
Frequency Range
a/
Loudnes
s Levels
(db)
U
84
64
43
24
-1
1000-2000
100
97
83
65
43
2
•Lcalc.
103
84
64
45
7
u
84
64
43
24
1
1000-2000
100
99
82
65
42
2
•^■cnlc.
103
84
64
45
11
u
79
60
40
20
10
-5
1150-2270
112
99
78
62
41
25
2
•"cole.
98
81
61
40
23
1
Lk
77
62
42
22
7
-7
1120-4520
340
102
86
66
46
20
2
J->ca\c.
101
88
69
44
19
-1
the data pertains to the loudness levels of the tones. Opposite Lk are
given the common loudness levels to which all the components of the
tone were adjusted for a particular test, and in the next line the results
of the test, that is, the observed loudness levels (L0bs.), are given.
Directly beneath each observed value, the calculated loudness levels
(Lc&ic) are shown. The three associated values of Lk, LohB., and
Lcau. in each column represent the data for one complete test. For
example, in Table VIII, the first tone is described as having ten com-
ponents, and for the first test shown each component was adjusted to
have a loudness level (Lk) of 67 db. The results of the test gave an
observed loudness level (Lohs.) of 83 db for the ten components acting
together, and the calculated loudness level (Z-caic.) of this same tone
was 81 db. The probable error of the observed results in the tables is
approximately ± 2 db.
TABLE X
Ten Component Tones (AL = 5 db)
Frequency Range
A/
Loudness Levels (db)
u
82
62
43
27
17
-6
1725-2220
55
101
73
54
38
30
2
Lenle.
95
76
56
40
30
-1
U
80
62
42
22
12
-2
1725-2220
55
94
66
50
33
22
2
•^ralc.
93
76
54
35
22
4
416
BELL SYSTEM TECHNICAL JOURNAL
In the next series of data, adjacent components had a difference in
loudness level of 5 db, that is, the first, third, fifth, etc., components
had the loudness level given opposite Lk, and the even numbered com-
ponents were 5 db lower. (Tables X and XI.)
TABLE XI
Eleven Component Tones (AZ, = 5 db).
Frequency Range
a/
Loudness Levels (db)
Lk
79
61
41
26
16
1
57-627
57
-Lobs.
91
73
56
41
28
2
-£<eulc.
90
76
59
43
28
8
Lk
76
61
42
25
15
-9
3420-4020
60
95
77
55
33
25
2
Lca\c.
89
75
54
36
26
-4
In the following set of tests (Tables XII and XIII) the difference in
loudness level of adjacent components was 10 db.
TABLE XII -
Ten Component Tones (AZ, = 10 db)
Frequency Range
A/
Loudness Levels
(db)
Lk
79
59
40
19
9
-5
1725-2220
55
95
71
54
33
22
2
•^calc.
91
73
51
31
17
-1
Lk
79
61
41
27
17
-1
1725-2220
55
89
67-
48
37
27
2
-Lcalc.
92
75
53
39
28
4
TABLE XIII
Eleven Component Tones (AZ = 10 db)
Frequency Range
A/
Loudness Levels
(db)
Lk
80
62
42
27
17
2
57-627
57
L0ba.
88
70
53
40
27
2
•^■calc.
90
76
59
45
30
8
Lk
81
62
42
27
17
-4
3420-4020
60
100
70
50
33
26
2
Lcalc.
94
75
53
37
27
0
The next data are the results of tests made on the complex tone
generated by the Western Electric No. 3A audiometer. When
LOUDNESS
417
analyzed, this tone was found to have the voltage level spectrum
shown in Table XIV. When the r.m.s. voltage across the receivers
used was unity, that is, zero voltage level, then the separate com-
ponents had the voltage levels given in this table. Adding to the
voltage levels the calibration constant for the receivers used in making
the loudness tests gives the values of /3 for zero voltage level across the
receivers. The values of /3 for any other voltage level are obtained by
addition of the level desired.
TABLE XIV
Voltage Level Spectrum of No. 3A Audiometer Tone
Frequency
Voltage Level
Frequency
Voltage Level
152
- 2.1
2128
-11.4
304
- 5.4
2280
-16.9
456
- 4.7
2432
-14.1
608
- 5.9
2584
-16.2
760
- 4.6
2736
-17.4
912
- 6.8
2880
-17.5
1064
- 6.0
3040
-20.0
1216
- 8.1
3192
-19.4
1368
- 7.6
3344
-22.7
1520
- 9.1
3496
-23.7
1672
-10.0
3648
-25.6
1824
- 9.9
3800
-24.6
1976
-14.1
3952
-26.8
Tests were made on the audiometer tone with the same receivers "
that were used with the other complex tones, but in addition, data were
available on tests made about six years ago using a different type of
receiver. This latter type of receiver was recalibrated (Fig. 13) and
computations made for both the old and new tests. In the older set
of data, levels above threshold were given instead of voltage levels,
so in utilizing it here, it was necessary to assume that the threshold
levels of the new and old tests were the same.
Computations were made at the levels tested experimentally and a
comparison of observed and calculated results is shown in Table XV.
The agreement of observed and calculated results is poor for some
of the tests, but the close agreement in the recent data at low levels
and in the previous data at high levels indicates that the observed
results are not as accurate as could be desired. Because of the labor
involved these tests have not been repeated.
At the time the tests were made several years ago on the No. 3A
Audiometer tone, the reduction in loudness level which takes place
when certain components are eliminated was also determined. As this
11 See Calibration shown in Fig. 1.
418
BELL SYSTEM TECHNICAL JOURNAL
100
95
90
85
80
75
\
\J
r\
\J
\
\
500 1000
FREQUENCY IN CYCLES PER SECOND
Fig. 13 — Calibration of receivers for tests on the No. 3A audiometer tone
can be readily calculated with the formula developed here, a com-
parison of observed and calculated results will be shown. In Fig. 14A,
the ordinate is the reduction in loudness level resulting when a No.
3A Audiometer tone having a loudness level of 42 db was changed by
the insertion of a filter which eliminated all of the components above
or below the frequency indicated on the abscissa. The observed data
are the plotted points and the smooth curves are calculated results.
A similar comparison is shown in Figs. 14B, C and D for other levels.
A.
TABLE XV
Recent Tests on No. 3A Audiometer Tone
R.m.s. Volt. Level
-38
-55
-59
-70
-75
-78
-80
-87
-89
-100
-102
■£>obs.
95
89
85
74
79
71
61
57
56
49
41
44
42
40
28
28
22
25
2
7
2
4
B.
Previous Tests on No. 3A Audiometer Tone
R.m.s. Volt. Level . . .
LobB.
■Lcalc.
118
119
103
103
77
82
69
73
61
56
50
41
-91
2
6
LOUDNESS
419
This completes the data which are available on steady complex
tones. It is to be hoped that others will find the field of sufficient im-
portance to warrant obtaining additional data for improving and
testing the method of measuring and calculating loudness levels.
In view of the complex nature of the problem this computation
method cannot be considered fully developed in all its details and as
more accurate data accumulates it may be necessary to change the
formula for b. Also at the higher levels some attention must be given
to phase differences between the components. However, we feel that
the form of the equation is fundamentally correct and the loudness
• -HIGH PASS x-LOW PASS
0
CD
? 5
1,0
u
2 15
a.
_i
UJ
>
Id
_i
in
z
§ 5
S
10
15
•
LOU
3NE
•
s
5 LEV
LI
EL = 42 0
B
•
•
S
-J
;<r
A
\
i
\
■
100
LOUDNESS LEV
EL = 87 DB
•
V
^x
'
<•
/
/
\
LOL
DN
ES
S
LE
/EL = 95
DB
•
•
^
**f
\«
i
m
\
1000 2000 4000
100
200 400
1000 2000 4000
D
c FILTER CUTOFF FREQUENCY
Fig. 14 (A to D) — Loudness level reduction tests on the No. 3A audiometer tone.
function, G, corresponds to something real in the mechanism of hearing.
The present values given for G may be modified slightly, but we think
that they will not be radically changed.
A study of the loudness of complex sounds which are not steady,
such as speech and sounds of varying duration, is in progress at the
present time and the results will be reported in a second paper on this
subject.
Appendix A. Experimental Method of Measuring the
Loudness Level of a Steady Sound
A measurement of the loudness level of a sound consists of listening
alternately to the sound and to the 1000-cycle reference tone and
adjusting the latter until the two are equally loud. If the intensity
420 BELL SYSTEM TECHNICAL JOURNAL
level of the reference tone is L decibels when this condition is reached,
the sound is said to have a loudness level of L decibels. When the
character of the sound being measured differs only slightly from that
of the reference tone, the comparison is easily and quickly made, but
for other sounds the numerous factors which enter into a judgment of
equality of loudness become important, and an experimental method
should be used which will yield results typical of the average normal
ear and normal physiological and psychological conditions.
A variety of methods have been proposed to accomplish this,
differing not only in general classification, that is, the method of
average error, constant stimuli, etc., but also in important experi-
mental details such as the control of noise conditions and fatigue
effects. In some instances unique devices have been used to facilitate
a ready comparison of sounds. One of these, the alternation
phonometer,12 introduces into the comparison important factors such
as the duration time of the sounds and the effect of transient condi-
tions. The merits of a particular method will depend upon the
circumstances under which it is to be used. The one to be described
here was developed for an extensive series of laboratory tests.
To determine when two sounds are equally loud it is necessary to
rely upon the judgment of an observer, and this involves of course,
not only the ear mechanism, but also associated mental processes, and
effectively imbeds the problem in a variety of psychological factors.
These difficulties are enhanced by the large variations found in the
judgments of different observers, necessitating an investigation con-
ducted on a statistical basis. The method of constant stimuli, wherein
the observer listens to fixed levels of the two sounds and estimates
which sound is the louder, seemed best adapted to control the many
factors involved, when using several observers simultaneously. By
means of this method, an observer's part in the test can be readily
limited to an indication of his loudness judgment. This is essential
as it was found that manipulation of apparatus controls, even though
they were not calibrated, or participation in any way other than as a
judge of loudness values, introduced undesirable factors which were
aggravated by continued use of the same observers over a long period
of time. Control of fatigue, memory effects, and the association of an
observer's judgments with the results of the tests or with the judgments
of other observers could be rigidly maintained with this method, as
will be seen from the detailed explanation of the experimental pro-
cedure.
12 D. Mackenzie, " Relative Sensitivity of the Ear at Different Levels of Loudness,"
Phys. Rev. 20, 331 (1922).
LOUDNESS
421
The circuit shown in Fig. 15 was employed to generate and control
the reference tone and the sounds to be measured. Vacuum tube
oscillators were used to generate pure tones, and for complex tones and
other sounds, suitable sources were substituted. By means of the
voltage measuring circuit and the attenuator, the voltage level
(voltage level = 20 log V) impressed upon the terminals of the re-
ceivers, could be determined. For example, the attenuator, which
was calibrated in decibels, was set so that the voltage measuring set
indicated 1 volt was being impressed upon the receiver. Then the
difference between this setting and any other setting is the voltage
level. To obtain the intensity level of the sound we must know the
calibration of the receivers.
FEEDBACK
CONTROL
X"
r^T" OSCLLATOR
' - r*1 iooo —
CRS.
ATTENUATOR
AMPLIFIER
FILTER
t®-
&&
MOTOR
THERMO-
COUPLE
CALIBRATED VOLTAGE
MEASURING CIRCUIT
o HEADPHONES
OSCILLATOR
X'
UwJU FR«UENCY
FEEDBACK
CONTROL
ATTENUATOR
AMPLIFIER
FILTER
Fig. 15 — Circuit for loudness balances.
The observers were seated in a sound-proof booth and were required
only to listen and then operate a simple switch. These switches were
provided at each position and were arranged so that the operations of
one observer could not be seen by another. This was necessary to
prevent the judgments of one observer from influencing those of
another observer. First they heard the sound being tested, and im-
mediately afterwards the reference tone, each for a period of one
second. After a pause of one second this sequence was repeated, and
then they were required to estimate whether the reference tone was
louder or softer than the other sound and indicate their opinions by
operating the switches. The levels were then changed and the pro-
cedure repeated. The results of the tests were recorded outside the
booth.
The typical recording chart shown in Fig. 16 contains the results of
three observers testing a 125-cycle tone at three different levels. Two
422 BELL SYSTEM TECHNICAL JOURNAL
125 c.p.s. Pure Tone Test No. 4 Crew No. 1. 1000 c.p.s. Voltage Level (db)
Obs.
+6
+2
-2
-6
-10
-14
-18
-22
-26
125
CK
+
+
+
+
+
0
0
0
0
c.p.s.
AS
+
+
+
+
0
0
0
0
0
Volt.
DH
+
+
0
0
0
0
0
0
0
level =
CK
+
+
+
+
+
0
0
0
0
+ 9.8 db
AS
+
+
+
+
0
0
0
0
0
DH
+
+
0
0
+
0
0
0
0
CK
+
+
+
+
0
0
0
0
0
AS
+
+
+
0
0
0
0
0
0
DH
+
+
0
0
0
0
0
0
0
0
-4
-8
-12
-16
-20
-24
-28
-32
125
CK
+
+
+
+
0
+
+
0
0
c.p.s.
AS
+
+
+
+
+
0
0
0
0
Volt.
DH
+
+
+
+
0
0
0
0
0
level =
CK
+
+
+
+
+
+
+
0
0
- 3.2 db
AS
+
+
+
+
+
+
0
0
0
DH
+
+
+
0
+
0
+
0
0
CK
+
+
+
+
+
+
0
0
0
AS
+
+
+
+
+
0
0
0
0
DH
+
+
+
0
+
0
0
0
0
-IS
-19
-23
-27
-31
-35
-39
-43
-47
125
CK
~T
+
+
+
+
0
0
0
0
c.p.s.
AS
+
+
+
+
0
0
0
0
0
Volt.
DH
+
+
0
+
0
+
0
0
0
level =
CK
0
+
+
+
+
+
0
0
0
- 14.2
AS
+
+
+
+
0
+
0
0
0
db
DH
+
+
0
+
0
0
+
0
0
CK
+
+
0
+
+
+
0
0
0
AS
+
+
0
0
+
+
0
0
0
DH
+
+
0
0
0
0
+
0
0
Fig. 16 — Loudness balance data sheet.
marks were used for recording the observers' judgments, a cipher
indicating the 125-cycle tone to be the louder, and a plus sign denoting
the reference tone to be the louder of the two. No equal judgments
were permitted. The figures at the head of each column give the
voltage level of the reference tone impressed upon the receivers, that is,
the number of decibels from 1 volt, plus if above and minus if below,
and those at the side are similar values for the tone being tested.
Successive tests were chosen at random from the twenty-seven possible
combinations of levels shown, thus reducing the possibility of memory
effects. The levels were selected so the observers listened to reference
tones which were louder and softer than the tone being tested and the
median of their judgments was taken as the point of equal loudness.
The data on this recording chart, combined with a similar number
LOUDNESS
423
of observations by the rest of the crew, (a total of eleven observers) are
shown in graphical form in Fig. 17. The arrow indicates the median
VOLT
AGE
-EVE
- = -3DB 1
s
\\
A
i
-18.6
/
/
VOLTAGE
LEVEL = -14 DB ,
<A
-3L0
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
PER CENT
Fig. 17 — Percent of observations estimating 1000-cycle tone to be louder than 125-
cycle tone.
level at which the 1000-cycle reference, in the opinion of this group of
observers, sounded equally loud to the 125-cycle tone.
The testing method adopted was influenced by efforts to minimize
fatigue effects, both mental and physical. Mental fatigue and
probable changes in the attitude of an observer during the progress of
a long series of tests were detected by keeping a record of the spread of
each observer's results. As long as the spread was normal it was
assumed that the fatigue, if present, was small. The tests were con-
ducted on a time schedule which limited the observers to five minutes
of continuous testing, during which time approximately fifteen obser-
vations were made. The maximum number of observations permitted
in one day was 150.
To avoid fatiguing the ear the sounds to which the observers listened
were of short duration and in the sequence illustrated on Fig. 18. The
WARNING
1
X
SOUND
1000
CP.S.
"X
SOUND
1000
CP.S.
BUZZ
1
1
1 I 1 i
1 l 1
I 1
0 0-5
IjO
1.5
2.0 2.5
3.0 3.5 4.0 4.5
TIME IN SECONDS
5.0 5.5 6.0
6.5 7.0
7.
Fig. 18 — Time sequence for loudness comparisons.
duration time of each sound had to be long enough to attain full loud-
ness and yet not sufficiently long to fatigue the ear. The reference
tone followed the x sound at a time interval short enough to permit a
424
BELL SYSTEM TECHNICAL JOURNAL
ready comparison, and yet not be subject to fatigue by prolonging
the stimulation without an adequate rest period. At high levels it
was found that a tone requires nearly 0.3 second to reach full loudness
and if sustained for longer periods than one second, there is danger of
fatiguing the ear.13
To avoid the objectionable transients which occur when sounds are
interrupted suddenly at high levels, the controlling circuit was de-
signed to start and stop the sounds gradually. Relays operating in
the feedback circuits of the vacuum tube oscillators and in the grid
circuits of amplifiers performed this operation. The period of growth
and decay was approximately 0. 1 second as shown on the typical
oscillogram in Fig. 19. With these devices the transient effects were
, ...mi!"! !!l| mil
■ , :
Growth
mnm 'nia^
":::x&
Decay
Fig. 19 — Growth and decay of 1000-cycle reference tone.
reduced and yet the sounds seemed to start and stop instantaneously
unless attention was called to the effect. A motor-driven commutator
operated the relays which started and stopped the sounds in proper
sequence, and switched the receivers from the reference tone circuit to
the sound under test.
13 G. v. Bekesy, "Theory of Hearing," Phys. Zeits. 30, 115 (1929).
LOUDNESS 425
The customary routine measurements to insure the proper voltage
levels impressed upon the receivers were made with the measuring
circuit shown schematically in Fig. 15. During the progress of the
tests voltage measurements were made frequently and later correlated
with measurements of the corresponding field sound pressures.
Threshold measurements were made before and after the loudness
tests. They were taken on the same circuit used for the loudness tests
(Fig. 15) by turning off the 1000-cycle oscillator and slowly attenuating
the other tone below threshold and then raising the level until it again
became audible. The observers signalled when they could no longer
hear the tone and then again when it was just audible. The average
of these two conditions was taken as the threshold.
An analysis of the harmonics generated by the receivers and other
apparatus was made to be sure of the purity of the tones reaching the
ear. The receivers were of the electrodynamic type and were found
to produce overtones of the order of 50 decibels below the fundamental.
At the very high levels, distortion from the filters was greater than
from the receivers, but in all cases the loudness level of any overtone
was 20 decibels or more below that of the fundamental. Experience
with complex tones has shown that under these conditions the con-
tribution of the overtones to the total loudness is insignificant.
The method of measuring loudness level which is described here has
been used on a large variety of sounds and found to give satisfactory
results.
Appendix B. Comparison of Data on the Loudness
Levels of Pure Tones
A comparison of the present loudness data with that reported
previously by B. A. Kingsbury 4 would be desirable and in the event
of agreement, would lend support to the general application of the
results as representative of the average ear. It will be remembered
that the observers listened to the tones with both ears in the tests
reported here, while a single receiver was used by Kingsbury.
Also, it is important to remember that the level of the tones used in
the experiments was expressed as the number of db above the average
threshold current obtained with a single receiver. For both of these
reasons a direct comparison of the results cannot be made. However,
in the course of our work two sets of experiments were made which
give results that make it possible to reduce Kingsbury's data so that
it may be compared directly with that reported in this paper.
In the first set of experiments it was found that if a typical ob-
server listened with both ears and estimated that two tones, the
426
BELL SYSTEM TECHNICAL JOURNAL
reference tone and a tone of different frequency, appeared equally
loud, then, making a similar comparison using one ear (the voltages
on the receiver remaining unchanged) he would still estimate that the
two tones were equally loud. The results upon which this conclusion
is based are shown in Table XVI. In the first row are shown the fre-
TABLE XVI
Comparison of One and Two-ear Loudness Balances
A. Reference tone voltage level = — 32 db
Frequency, c.p.s.
Voltage level
difference *
62
125
250
500
2000
4000
6000
8000
-0.5
0
+ 1.0
-1.0
-0.5
-0.5
+0.5
-3.0
10,000
-3.0
B. Other reference tone levels
62
c.p.s.
2000
c.p.s.
Ref. Tone Volt.
Level
Volt. Level Dif-
ference *
Ref. Tone Volt.
Level
Volt. Level Dif-
ference *
-20
-34
-57
-68
+0.5
+0.2
+ 2.0
-0.5
- 3
-22
-41
-60
-79
0.0
+0.3
-0.8
-0.8
-6.2
* Differences are in db, positive values indicating a higher voltage for the one ear
balance.
quencies of the tones tested. Under these frequencies are shown the
differences in db of the voltage levels on the receivers obtained when
listening by the two methods, the voltage level of the reference tone
being constant at 32 db down from 1 volt. Under the caption "Other
Reference Tone Levels" similar figures for frequencies of 62 c.p.s. and
2000 c.p.s. and for the levels of the reference tone indicated are given.
It will be seen that these differences are well within the observational
error. Consequently, the conclusion mentioned above seems to be
justified. This is an important conclusion and although the data are
confined to tests made with receivers on the ear it would be expected
that a similar relation would hold when the sounds are coming directly
to the ears from a free wave.
This result is in agreement with the point of view adopted in de-
veloping the formula for calculating loudness. When listening with
one instead of two ears, the loudness of the reference tone and also
that of the tone being compared are reduced to one half. Conse-
quently, if they were equally loud when listening with two ears they
must be equally loud when listening with one ear. The second set of
LOUDNESS
427
data is concerned with differences in the threshold when listening with
one ear versus listening with two ears.
It is well known that for any individual the two ears have different
acuity. Consequently, when listening with both ears the threshold is
determined principally by the better ear. The curve in Fig. 20 shows
the difference in the threshold level between the average of the better
of an observer's ears and the average of all the ears. The circles
represent data taken on the observers used in our loudness tests while
the crosses represent data taken from an analysis of 80 audiograms of
persons with normal hearing. If the difference in acuity when listening
with one ear vs. listening with two ears is determined entirely by the
better ear, then the curve shown gives this difference. However,
some experimental tests which we made on one ear acuity vs. two ear
acuity showed the latter to be slightly greater than for the better ear
alone, but the small magnitudes involved and the difficulty of avoiding
id 6
200 500 1000 2000
FREQUENCY IN CYCLES PER SECOND
10000 20000
Fig. 20 — Difference in acuity between the best ear and the average of both ears.
psychological effects caused a probable error of the same order of
magnitude as the quality being measured. At the higher frequencies
where large differences are usually present the acuity is determined
entirely by the better ear.
From values of the loudness function G, one can readily calculate
what the difference in acuity when using one vs. two ears should be.
Such a calculation indicates that when the two ears have the same
acuity, then when listening with both ears the threshold values are
about 2 db lower than when listening with one ear. This small
difference would account for the difficulty in trying to measure it.
We are now in a position to compare the data of Kingsbury with
those shown in Table I. The data in Table I can be converted into
decibels above threshold by subtracting the average threshold value in
each column from any other number in the same column.
If now we add to the values for the level above threshold given by
428
BELL SYSTEM TECHNICAL JOURNAL
Kingsbury an amount corresponding to the differences shown by the
curve of Fig. 20, then the resulting values should be directly com-
parable to our data on the basis of decibels above threshold. Com-
parisons of his data on this basis with those reported in this paper are
shown in Fig. 21. The solid contour lines are drawn through points
taken from Table I and the dotted contour lines taken from Kings-
bury's data. It will be seen that the two sets of data are in good
agreement between 100 and 2000 cycles but diverge somewhat above
and below these points. The discrepancies are slightly greater than
would be expected from experimental errors, but might be explained
500 1000
FREQUENCY IN CYCLES PER SECOND
Fig. 21 — Loudness levels of pure tones — A comparison with Kingsbury's data.
by the presence of a slight amount of noise during threshold de-
terminations.
Appendix C. Optical Tone Generator of Complex Wave Forms
For the loudness tests in which the reference tone was compared
with a complex tone having components of specified loudness levels
and frequencies, the tones were listened to by means of head receivers
as before; the circuit shown in Fig. 15 remaining the same excepting
for the vacuum tube oscillator marked "x Frequency." This was
replaced by a complex tone generator devised by E. C. Wente of the
Bell Telephone Laboratories. The generator is shown schematically
in Fig. 22.
LOUDNESS
429
r
A
^
_£
MOTOR
Fig. 22 — Schematic of optical tone generator.
The desired wave form was accurately drawn on a large scale and
then transferred photographically to the glass disk designated as D in
the diagram. The disk, driven by a motor, rotated between the lamp
L and a photoelectric cell C, producing a fluctuating light source which
Fig. 23 — Ten disk optical tone generator.
430 BELL SYSTEM TECHNICAL JOURNAL
was directed by a suitable optical system upon the plate of the cell.
The voltage generated was amplified and attenuated as in the case of
the pure tones.
The relative magnitudes of the components were of course fixed by
the form of the wave inscribed upon the disk, but this was modified
when desired, by the insertion of elements in the electrical circuit which
gave the desired characteristic. Greater flexibility in the control of
the amplitude of the components was obtained by inscribing each
component on a separate disk with a complete optical system and
cell for each. Frequency and phase relations were maintained by
mounting all of the disks on a single shaft. Such a generator having
ten disks is shown in Fig. 23.
An analysis of the voltage output of the optical tone generators
showed an average error for the amplitude of the components of about
±0.5 db, which was probably the limit of accuracy of the measuring
instrument. Undesired harmonics due to the disk being off center or
inaccuracies in the wave form were removed by filters in the electrical
circuit.
All of the tests on complex tones described in this paper were made
with the optical tone generator excepting the audiometer, and two
tone tests. For the latter tests, two Vacuum tube oscillators were
used as a source.
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